We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from [5], but have no rigorously proven convergence theory. Therefore we show how the monotonicity methods can be converted into a regularization method for a data-fitting functional without losing the convergence properties of the monotonicity methods. This is a great advantage and a significant improvement over standard regularization techniques. In more detail, we introduce constraints on the minimization problem of the residual based on the monotonicity methods and prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. In addition, we compare numerical reconstructions of inclusions based on the monotonicity-based regularization with a standard approach (one-step linearization with Tikhonov-like regularization), which also shows the robustness of our method regarding noise in practice.
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The time-dependent quadratic minimization (TDQM) problem appears in many applications and research projects. It has been reported that the zeroing neural network (ZNN) models can effectively solve the TDQM problem. However, the convergent and robust performance of the existing ZNN models are restricted for lack of a joint-action mechanism of adaptive coefficient and integration enhanced term. Consequently, the residual-based adaption coefficient zeroing neural network (RACZNN) model with integration term is proposed in this paper for solving the TDQM problem. The adaptive coefficient is proposed to improve the performance of convergence and the integration term is embedded to ensure the RACZNN model can maintain reliable robustness while perturbed by variant measurement noises. Compared with the state-of-the-art models, the proposed RACZNN model owns faster convergence and more reliable robustness. Then, theorems are provided to prove the convergence of the RACZNN model. Finally, corresponding quantitative numerical experiments are designed and performed in this paper to verify the performance of the proposed RACZNN model.
Nonlinear model order reduction has opened the door to parameter optimization and uncertainty quantification in complex physics problems governed by nonlinear equations. In particular, the computational cost of solving these equations can be reduced by means of local reduced-order bases. This article examines the benefits of a physics-informed cluster analysis for the construction of cluster-specific reduced-order bases. We illustrate that the choice of the dissimilarity measure for clustering is fundamental and highly affects the performances of the local reduced-order bases. It is shown that clustering with an angle-based dissimilarity on simulation data efficiently decreases the intra-cluster Kolmogorov $N$-width. Additionally, an a priori efficiency criterion is introduced to assess the relevance of a ROM-net, a methodology for the reduction of nonlinear physics problems introduced in our previous work in [T. Daniel, F. Casenave, N. Akkari, D. Ryckelynck, Model order reduction assisted by deep neural networks (ROM-net), Advanced Modeling and Simulation in Engineering Sciences 7 (16), 2020]. This criterion also provides engineers with a very practical method for ROM-nets' hyperparameters calibration under constrained computational costs for the training phase. On five different physics problems, our physics-informed clustering strategy significantly outperforms classic strategies for the construction of local reduced-order bases in terms of projection errors.
In this paper, we formulate and study substructuring type algorithm for the Cahn-Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature and since then it has appeared in many fields ranging from tumour growth simulation, image processing, thin liquid films, population dynamics etc. Being a non-linear equation, it is important to develop robust numerical techniques to solve the CH equation. Here we present the formulation of Dirichlet-Neumann (DN) and Neumann-Neumann (NN) methods applied to CH equation and study their convergence behaviour. We consider the domain-decomposition based DN and NN methods in one and two space dimension for two subdomains and extend the study for multi-subdomain setting for NN method. We verify our findings with numerical results.
Neural networks have shown tremendous growth in recent years to solve numerous problems. Various types of neural networks have been introduced to deal with different types of problems. However, the main goal of any neural network is to transform the non-linearly separable input data into more linearly separable abstract features using a hierarchy of layers. These layers are combinations of linear and nonlinear functions. The most popular and common non-linearity layers are activation functions (AFs), such as Logistic Sigmoid, Tanh, ReLU, ELU, Swish and Mish. In this paper, a comprehensive overview and survey is presented for AFs in neural networks for deep learning. Different classes of AFs such as Logistic Sigmoid and Tanh based, ReLU based, ELU based, and Learning based are covered. Several characteristics of AFs such as output range, monotonicity, and smoothness are also pointed out. A performance comparison is also performed among 18 state-of-the-art AFs with different networks on different types of data. The insights of AFs are presented to benefit the researchers for doing further research and practitioners to select among different choices. The code used for experimental comparison is released at: \url{//github.com/shivram1987/ActivationFunctions}.
Despite their overwhelming capacity to overfit, deep neural networks trained by specific optimization algorithms tend to generalize well to unseen data. Recently, researchers explained it by investigating the implicit regularization effect of optimization algorithms. A remarkable progress is the work (Lyu&Li, 2019), which proves gradient descent (GD) maximizes the margin of homogeneous deep neural networks. Except GD, adaptive algorithms such as AdaGrad, RMSProp and Adam are popular owing to their rapid training process. However, theoretical guarantee for the generalization of adaptive optimization algorithms is still lacking. In this paper, we study the implicit regularization of adaptive optimization algorithms when they are optimizing the logistic loss on homogeneous deep neural networks. We prove that adaptive algorithms that adopt exponential moving average strategy in conditioner (such as Adam and RMSProp) can maximize the margin of the neural network, while AdaGrad that directly sums historical squared gradients in conditioner can not. It indicates superiority on generalization of exponential moving average strategy in the design of the conditioner. Technically, we provide a unified framework to analyze convergent direction of adaptive optimization algorithms by constructing novel adaptive gradient flow and surrogate margin. Our experiments can well support the theoretical findings on convergent direction of adaptive optimization algorithms.
In this paper, we propose a novel Feature Decomposition and Reconstruction Learning (FDRL) method for effective facial expression recognition. We view the expression information as the combination of the shared information (expression similarities) across different expressions and the unique information (expression-specific variations) for each expression. More specifically, FDRL mainly consists of two crucial networks: a Feature Decomposition Network (FDN) and a Feature Reconstruction Network (FRN). In particular, FDN first decomposes the basic features extracted from a backbone network into a set of facial action-aware latent features to model expression similarities. Then, FRN captures the intra-feature and inter-feature relationships for latent features to characterize expression-specific variations, and reconstructs the expression feature. To this end, two modules including an intra-feature relation modeling module and an inter-feature relation modeling module are developed in FRN. Experimental results on both the in-the-lab databases (including CK+, MMI, and Oulu-CASIA) and the in-the-wild databases (including RAF-DB and SFEW) show that the proposed FDRL method consistently achieves higher recognition accuracy than several state-of-the-art methods. This clearly highlights the benefit of feature decomposition and reconstruction for classifying expressions.
Vision-based Simultaneous Localization And Mapping (VSLAM) is a mature problem in Robotics. Most VSLAM systems are feature based methods, which are robust and present high accuracy, but yield sparse maps with limited application for further navigation tasks. Most recently, direct methods which operate directly on image intensity have been introduced, capable of reconstructing richer maps at the cost of higher processing power. In this work, an edge-based monocular SLAM system (SE-SLAM) is proposed as a middle point: edges present good localization as point features, while enabling a structural semidense map reconstruction. However, edges are not easy to associate, track and optimize over time, as they lack descriptors and biunivocal correspondence, unlike point features. To tackle these issues, this paper presents a method to match edges between frames in a consistent manner; a feasible strategy to solve the optimization problem, since its size rapidly increases when working with edges; and the use of non-linear optimization techniques. The resulting system achieves comparable precision to state of the art feature-based and dense/semi-dense systems, while inherently building a structural semi-dense reconstruction of the environment, providing relevant structure data for further navigation algorithms. To achieve such accuracy, state of the art non-linear optimization is needed, over a continuous feed of 10000 edgepoints per frame, to optimize the full semi-dense output. Despite its heavy processing requirements, the system achieves near to real-time operation, thanks to a custom built solver and parallelization of its key stages. In order to encourage further development of edge-based SLAM systems, SE-SLAM source code will be released as open source.
With the advent of deep neural networks, learning-based approaches for 3D reconstruction have gained popularity. However, unlike for images, in 3D there is no canonical representation which is both computationally and memory efficient yet allows for representing high-resolution geometry of arbitrary topology. Many of the state-of-the-art learning-based 3D reconstruction approaches can hence only represent very coarse 3D geometry or are limited to a restricted domain. In this paper, we propose occupancy networks, a new representation for learning-based 3D reconstruction methods. Occupancy networks implicitly represent the 3D surface as the continuous decision boundary of a deep neural network classifier. In contrast to existing approaches, our representation encodes a description of the 3D output at infinite resolution without excessive memory footprint. We validate that our representation can efficiently encode 3D structure and can be inferred from various kinds of input. Our experiments demonstrate competitive results, both qualitatively and quantitatively, for the challenging tasks of 3D reconstruction from single images, noisy point clouds and coarse discrete voxel grids. We believe that occupancy networks will become a useful tool in a wide variety of learning-based 3D tasks.
We propose an algorithm for real-time 6DOF pose tracking of rigid 3D objects using a monocular RGB camera. The key idea is to derive a region-based cost function using temporally consistent local color histograms. While such region-based cost functions are commonly optimized using first-order gradient descent techniques, we systematically derive a Gauss-Newton optimization scheme which gives rise to drastically faster convergence and highly accurate and robust tracking performance. We furthermore propose a novel complex dataset dedicated for the task of monocular object pose tracking and make it publicly available to the community. To our knowledge, It is the first to address the common and important scenario in which both the camera as well as the objects are moving simultaneously in cluttered scenes. In numerous experiments - including our own proposed data set - we demonstrate that the proposed Gauss-Newton approach outperforms existing approaches, in particular in the presence of cluttered backgrounds, heterogeneous objects and partial occlusions.
TraQuad is an autonomous tracking quadcopter capable of tracking any moving (or static) object like cars, humans, other drones or any other object on-the-go. This article describes the applications and advantages of TraQuad and the reduction in cost (to about 250$) that has been achieved so far using the hardware and software capabilities and our custom algorithms wherever needed. This description is backed by strong data and the research analyses which have been drawn out of extant information or conducted on own when necessary. This also describes the development of completely autonomous (even GPS is optional) low-cost drone which can act as a major platform for further developments in automation, transportation, reconnaissance and more. We describe our ROS Gazebo simulator and our STATUS algorithms which form the core of our development of our object tracking drone for generic purposes.
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